I smell a rat.
It unlikely that a collection agency will contact you by phone when you haven't received anything in the mail. The Fair Debt Collection Act (something like that) says they have to notify you in writing and you have a certain amount of time to dispute the validity of the debt among other things.
I don't see why anyone would ask you to verify your entire social security number over the phone. Usually it's only the last 4 digits. The fact that they said your full name then your full social security number and you verified it (shame on you, you know better) is a red flag. Notice how quickly they hung up after verifying your social?
While some collections agencies will sue for small amounts, the majority of them don't. To answer someone else, yes you have to be served with papers notifying you an action has been filed against you, you have X number of days to respond to the petition, etc. Threats of legal action by a collection agency is just that, threats.
It's not hard to get an account with a third party business that allows you to pull other peoples credit reports to start running a scam. All you have to do is pay the per report fee, or a flat fee depending on the company you use.
If I were you this is what I would do and worry about the validity later. That verifying the whole social over the phone scares me.
Call your phone carrier and ask if they have any type of reverse call blocking where you can block the last number that called your phone.
Go to
www.annualcreditreport.com and pull a credit report from each of the agencies, there will be 3. These reports are free and you get one free report per year from each agency.
Once you have reviewed all 3 reports (and printed them) contact the agencies and have them put a "fraud alert" on your account. This will prevent anyone using or stealing your identity.
Look through all your medical bills and see if you have one for $600. If you don't, bigger red flag. If you do, call them and find out what is going on.